Tangent spaces and derivatives

Partial derivatives with respect to local coordinates

Let $U\to\mathbf{R}^n$ be a smooth chart defined over an open set $U$ in $M$. Then there are smooth real-valued functions $x^1,x^2,\dots,x^n$ on $U$ such that

$\displaystyle \varphi(u)=(x^1(u),x^2(u),\dots,x^n(u))$

for all $u\in U$. The functions $x^1,x^2,\dots,x^n$ determined by the chart $(U,\varphi)$ constitute smooth local coordinate functions defined over the domain $U$ of the chart.

Definition Let $x^1,x^2,\dots,x^n$ be smooth local coordinates defined over an open set $U$ in a smooth manifold $M$ of dimension $n$, and let $V$ be the corresponding open set in $\mathbf{R}^n$ defined such that